Elimination Method
How to solve a system of linear equations and inequalities by using their graphs: 4 examples and their solutions.
Example 1
Example
Solution
Choose the variable you want to remove.
The same or the opposite terms are easier to remove.
So choose y as the variable to remove.
Make the y terms the same or the opposites.
-y and +y are already the opposites.
So write x - y = 4 and 2x + y = 5.
To remove the y terms,
add the equations.
x + 2x = 3x
The y terms are removed.
4 + 5 = 9
So 3x = 9.
Divide both sides by 3.
Then x = 3.
Put this x = 3
into one of the given equations.
x - y = 4 looks simpler.
So put x = 3 into x - y = 4.
Then 3 - y = 4.
Substitution Method
Find the value of y.
Then y = -1.
So [x = 3, y = -1] is the answer.
Example 2
Example
Solution
Choose the variable you want to remove.
This time, let's choose the variable x to remove.
Make the x terms the same or the opposites.
The x terms are 3x and 2x.
So change 3x and 2x to 6x.
So multiply 2 to the both sides of 3x + 2y = 7:
6x + 4y = 14.
Next, see the other equation 2x - 3y = -4.
To change 2x to 6x,
multiply 3 to the both sides of 2x - 3y = -4:
6x - 9y = -12.
So the given equations are changed to
6x + 4y = 14 and 6x - 9y = -12.
The x terms are the same: 6x.
To remove the x terms,
subtract the equations.
The x terms are removed.
4y - (-9y) = 4y + 9y = 13y
14 - (-12) = 14 + 12 = 26
So 13y = 26.
Divide both sides by 13.
Then y = 2.
Put this y = 2
into one of the given equations.
Put y = 2 into 3x + 2y = 7.
Then 3x + 2⋅2 = 7.
Find the value of x.
Then x = 1.
Linear Equation (One Variable)
So [x = 1, y = 2] is the answer.
Example 3
Example
Solution
Choose the variable you want to remove.
Choose x as the variable to remove.
Make the x terms the same or the opposites.
The x terms are x and 2x.
So, to make the same 2x,
change x to 2x.
So multiply 2 to the both terms of x - y = 4:
2x - 2y = 8.
And write 2x - 2y = 8.
To remove the x terms,
subtract the equations.
The x terms are removed.
The y terms are removed.
8 - 8 = 0
So 0 = 0.
0 = 0 is always true.
Just like this case,
when you get an equation that is always true,
then the system has infinitely many solutions.
So [infinitely many solutions] is the answer.
Example 4
Example
Solution
Choose the variable you want to remove.
Choose x as the variable to remove.
Make the x terms the same or the opposites.
The x terms are both the same x.
So write x - y = 4 and x - y = -3.
To remove the x terms,
subtract the equations.
The x terms are removed.
The y terms are removed.
4 - (-3) = 4 + 3 = 7
So 0 = 7.
0 = -7 is always false.
Just like this case,
when you get an equation that is always false,
then the system has no solution.
So [no solution] is the answer.